| Abstract: |
| Mixed-mode oscillations consisting of alternating small- and large-amplitude oscillations are often caused by folded singularities and singular Hopf bifurcations. We show that coupling between identical nonlinear oscillators can cause mixed-mode oscillations because of symmetry breaking. This behavior is illustrated for FitzHugh-Nagumo oscillators with repulsive coupling, and we show that it is caused by a singular Hopf bifurcation related to singularities at a cusp -- not a fold -- of the critical manifold.
Using blowup, we determine the number of small-amplitude oscillations analytically, showing -- as for the folded nodes -- that they are determined by the ratio of eigenvalues. The model undergoes a saddle-node bifurcation in the desingularized reduced problem, which also occurs on a cusp, and not a fold. We find excellent agreement between our analytical results and numerical computations.
Inspired by earlier work, we then identify a new type of bursting dynamics in a model of two coupled beta-cells. Small-amplitude oscillations in the action potential height precede transitions to square-wave bursting. This behavior is related to a pitchfork-of-limit-cycles bifurcation in the fast subsystem caused by symmetry breaking. Moreover, we show that it is organized by a cusped saddle-node in the averaged system as seen for the coupled FitzHugh-Nagumo units. |
|